Post on 02-Jun-2018
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8/10/2019 Ce 226 11415 Part 04 - Equiv Viscous Damping
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The energy input to an SDOF by imparting to it an initialdisplacement and initial velocity is given by
Recall he displacement response of a free vibration with initial displacement and velocity:
2.3 Energy in Free Vibration
This is the same as the input energy in Eq. 2.3.1. Therefore the total energy is
independent of time and is dependent only on the input energy. Hence, there is
conservation of energy during free vibration of a system without damping.
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The energy dissipated is proportional to the square of the amplitude of motion. It
is not a constant value for any given amount of damping and amplitude since the
energy dissipated increases linearly with the excitation frequency.
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In steady-state vibration, the energy input to the system due to the applied
force is dissipated in viscous damping. The external forcep(t) inputs energy tothe system which for each cycle is
Using Eq. 3.2.12:
Eq. 3.8.2 can be written as (see Derivation 3.6)
Hence, from Eq. 3.8.1 and 3.8.3, ED = EI
The preceding energy concepts help explain the growth of thedisplacement amplitude caused by harmonic force with = nuntil steady-state is attained as shown in Fig 3.2.2. For = n ,=90o and Eq. 3.8.2 gives
EI= po uo (3.8.4)
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. 3.8.2 ()
()
()
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, .., ,fl , .,
.
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(2)
damping ratio and damping coefficient c.
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.
() / 390 / 0.15 2600 /
() . 3..2
2
26
, 0.071 7.1%
/ 2 / 2 ( )
(1/4)(/)
(2 ) 2 / 2(0.071)(2600)/10 36.92 /